## Problems for Cumulative Distribution Function

Earlier we discussed the introduction of a continuous random variable and cumulative distribution function. Please see this article if you haven’t read it about already. Now we will give some problems related to Cumulative distribution function so you can have a deeper understanding of this topic. These problems are adapted from the following textbook: “Probability and Random Processes by Scott Miller 2nd Edition.” Problem-1: Which of the following mathematical functions could be the CDF of some random variable? Remember: To be a CDF of some random variable, the function $$F_{X}(x)$$ must start at zero when $$x = -\infty$$, end at one...

## Continuous Random Variables and Cumulative Distribution Function

The problem with Discrete random variables was they can’t be used for continuous data. In engineering, there comes a time when we need to model continuous random data, and it happens a lot the most popular random variable which we use is Gaussian Random Variable. Since Discrete Random Variables can be characterized or fully described by their PMF (Probability Mass Function) we can’t use the same for a continuous case because the probability would be 0. We can explain this by giving an example. To see discrete random variables please visit this article. Suppose you want to measure a temperature of...

## Practice Problems (Introduction to Probability Theory)

A few practice problems related to Probability theory are given here. These problems are adapted from the textbook: Probability and Random Processes by Scott Miller 2nd Edition. These problems cover the following topics: Experiments, Sample spaces, events, Axioms of Probability, Assigning Probability, Joint and Conditional Probability, Independence, Baye’s Theorem, Total Law of Probability, Discrete Random variables. Problem-1: A roulette wheel consists of 38 numbers (18 are red, 18 are black, and 2 are green). Assume that with each spin of the wheel, each number is equally likely to appear. (a) What is the probability of a gambler winning if he...

## Random Variables

A random variable is a real valued function of the elements of sample space S which maps each possible outcome  to a real number specified by some rule. Random variables are denoted by capital Alphabet like X. If the random variable takes on finite or countably an infinite number of values then the random variable is a discrete random variable and if it takes uncountably infinite values then it is a continuous random variable. As we said earlier that the mapping is done by some rule; Probability mass function $$P_X(x)$$ defines that rule. $$P_X(x)$$ of a random variable is a function that...

## Baye’s Rule

In Probability theory, we came across some problems in which it is difficult to compute conditional probability directly. To explain this let us consider two events A and B, then their joint probability is given as: $$\begin{eqnarray}P(A,B) = P(A|B)P(B) \label{eqA}\end{eqnarray}$$ and we know that: $$\begin{eqnarray}P(B|A) = \frac{P(A,B)}{P(A)} \label{eqB}\end{eqnarray}$$ then using \eqref{eqA} and \eqref{eqB} we get: $$\begin{eqnarray}P(B|A) = \frac{P(A|B)P(B)}{P(A)} \label{bayes}\end{eqnarray}$$ Equation \eqref{bayes} is called Baye’s Rule and is useful for calculating certain conditional probabilities as in many problems it is difficult to compute P(B|A) directly, hence we use Baye’s Rule to solve such problems. A few examples are given below:...